{
  "abstract": "Keith numbers are integers that reappear in sliding-window recurrences generated from their own digits. We introduce Keith mirrors, a subclass defined by saturation at the center of the recurrence's incidence profile. For an even index m, let C_d(m) denote the sum of the two central incidence counts. We prove that the central deficit m-C_d(m) is exactly the distance from m to the interval [2d,2d+2], which forces every Keith mirror of order d>=3 in base B>=3 to return at 2d+2. A digit identity and analytic capacity bounds then reduce the classification to finitely many cases. Exact enumeration, independently checked against the original recurrence and incidence definition, yields precisely 35 mirrors of order at least three, while order two consists of two explicit infinite families. We also classify exactly eight cross-base triple alignments satisfying R=C=x_1+x_d and record the conditional reconstruction property induced by incidence symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10557",
  "author_total": 1,
  "authors": [
    {
      "name": "Federico Ignacio Zamponi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2qzfh79nvezf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10557",
  "fulltext_url": null,
  "id": "c2qzfh79nvezf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10557",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Theories"
    ]
  ],
  "openalex": "W7221075263",
  "pdf_url": "https://arxiv.org/pdf/2610.10557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2qzfh79nvezf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10557"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Advanced Mathematical Theories",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The Mirror Theorem: Central Incidence Saturation in Keith Recurrences",
  "title_text": "The Mirror Theorem: Central Incidence Saturation in Keith Recurrences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/c2qzfh79nvezf",
  "venue": "arXiv",
  "versions": []
}